Numerical study of incommensurate and decoupled phases of spin-1/2 chains with isotropic exchange J1, J2 between first and second neighbors

Abstract

The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1 and has a singlet ground state (GS) for J1/J2 ⩾ −4. Its rich quantum phase diagram supports gapless, gapped, commensurate (C), incommensurate (IC) and other phases. Critical points J1/J2 are evaluated using exact diagonalization and density matrix renormalization group calculations. The wave vector qG of spin correlations is related to GS degeneracy and obtained as the peak of the spin structure factor S(q). Variable qG indicates IC phases in two J1/J2 intervals, [−4, − 1.24] and [0.44, 2], and a C–IC point at J1/J2 = 2. The decoupled C phase in [−1.24, 0.44] has constant qG = π/2, nondegenerate GS, and a lowest triplet state with broken spin density on sublattices of odd and even numbered sites. The lowest triplet and singlet excitations, Em and Eσ, are degenerate in finite systems at specific frustration J1/J2. Level crossing extrapolates in the thermodynamic limit to the same critical points as qG. The S(q) peak diverges at qG = π in the gapless phase with J1/J2 > 4.148 and quasi-long-range order (QLRO(π)). S(q) diverges at ±π/2 in the decoupled phase with QLRO(π/2), but is finite in gapped phases with finite-range correlations. Numerical results and field theory agree at small J2/J1 but disagree for the decoupled phase with weak exchange J1 between sublattices. Two related models are summarized: one has an exact gapless decoupled phase with QLRO(π/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.